Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #429476 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,18)(5,24)(6,14)(7,12)(8,13)(9,16)(10,22)(15,30)(17,26)(19,32)(20,27)(21,29)(23,28)(25,31);(1,3,6,17,8,16,7,5)(2,9,12,26,13,11,14,24)(4,19,22,31,21,15,23,20)(10,25,18,32,28,27,29,30);(1,4)(2,10)(3,15)(5,20)(6,23)(7,22)(8,21)(9,25)(11,27)(12,29)(13,28)(14,18)(16,19)(17,31)(24,32)(26,30);(1,5,6,16,8,17,7,3)(2,11,12,26,13,9,14,24)(4,20,23,19,21,31,22,15)(10,27,29,30,28,25,18,32)Polycyclic presentation:
2^9|b^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72219485, #72219488, #72219494, … |
| maximal normal subgroup of | #72219485, #72219488, #72219494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336672);